The next time a core-collapse supernova (SN) explodes in our galaxy, various detectors will be ready and waiting to detect its emissions of gravitational waves (GWs) and neutrinos. Current numerical simulations have successfully introduced multi-dimensional effects to produce exploding SN models, but thus far the explosion mechanism is not well understood. In this paper, we focus on an investigation of progenitor core rotation via comparison of the start time of GW emission and that of the neutronization burst. The GW and neutrino detectors are assumed to be, respectively, the KAGRA detector and a co-located gadolinium-loaded water Cherenkov detector, either EGADSmore » or GADZOOKS!. Our detection simulation studies show that for a nearby SN (0.2 kpc) we can confirm the lack of core rotation close to 100% of the time, and the presence of core rotation about 90% of the time. Using this approach there is also the potential to confirm rotation for considerably more distant Milky Way SN explosions.« less